A Maximal Large Deviation Inequality for Sub-Gaussian Variables

Dotan Di Castro; Claudio Gentile; Shie Mannor

A Maximal Large Deviation Inequality for Sub-Gaussian Variables

Machine Learning 2011-07-26 v3

Authors: Dotan Di Castro , Claudio Gentile , Shie Mannor

Abstract

In this short note we prove a maximal concentration lemma for sub-Gaussian random variables stating that for independent sub-Gaussian random variables we have $P<(\max_{1\le i\le N}S_{i}>\epsilon>) \le\exp<(-\frac{1}{N^2}\sum_{i=1}^{N}\frac{\epsilon^{2}}{2\sigma_{i}^{2}}>),$ where $S_i$ is the sum of $i$ zero mean independent sub-Gaussian random variables and $\sigma_i$ is the variance of the $i$ th random variable.

Keywords

gaussian estimation

Cite

@article{arxiv.1105.2550,
  title  = {A Maximal Large Deviation Inequality for Sub-Gaussian Variables},
  author = {Dotan Di Castro and Claudio Gentile and Shie Mannor},
  journal= {arXiv preprint arXiv:1105.2550},
  year   = {2011}
}

Comments

This paper has been withdrawn by the authors due to a crucial error in the last sentence of the proof of Theorem 1: "we can take the infimum of the r.h.s. over s, which yields (1)." This statement is only true if a single value of s yields the supremum of (\epsilon_i s - \rho_i(s)) simultaneously for every i

A Maximal Large Deviation Inequality for Sub-Gaussian Variables

Abstract

Keywords

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